Math Problem Statement
give a simple solution for all and with understandable explanation
Solution
Problem Breakdown and Solution:
The function given is , where .
We need to answer two questions:
- (i) Determine the image set of under and conclude whether is surjective.
- (ii) Solve for such that and conclude whether is injective.
Part (i): Image Set and Surjectivity
Step 1: Behavior of the sine function
The function has the following properties:
- The range (or image) of is because the sine function only takes values in this interval.
- Even though (all real numbers), the output of is always between and .
Step 2: Surjectivity
A function is surjective (onto) if every element in (codomain) has a preimage in .
Here:
- The codomain is (all real numbers).
- The range (actual output values) of is only .
Since the codomain includes all real numbers, but does not cover values outside , the function is not surjective.
Part (ii): Solving and Injectivity
Step 1: Solving
The equation can be solved as follows:
- The sine function equals at specific angles:
- This simplifies to: Here, represents all integers because the sine function is periodic with period .
Step 2: Injectivity
A function is injective (one-to-one) if different inputs and result in different outputs: .
Here:
- The sine function is periodic, meaning it repeats its values every .
- For , there are infinitely many values (e.g., , , and their periodic extensions).
Since multiple inputs give the same output, is not injective.
Final Answers:
- The image set of is , and is not surjective.
- The equation has multiple solutions, so is not injective.
Clarification Questions or Follow-ups:
- What is the domain and range of trigonometric functions?
- Why is the sine function periodic?
- How do you determine surjectivity and injectivity?
- Can a function be injective but not surjective?
- What are the general solutions for ?
Tip: To check surjectivity and injectivity, always compare the range of the function with its codomain and analyze for repeated inputs.
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Math Problem Analysis
Mathematical Concepts
Functions
Surjectivity
Injectivity
Trigonometry
Sine Function
Formulas
f(x) = sin(x)
General solutions for sin(x) = k
Theorems
Definition of surjective functions
Definition of injective functions
Periodicity of sine function
Suitable Grade Level
Grades 11-12
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